(1) Technical Field
The present invention relates to techniques for system simulations. More specifically, the present invention relates to a technique for using wavelet operator to simulate the performance of mixed-signal circuits.
(2) Discussion
Mixed-signal circuits, an example of which is the delta-sigma modulator, include both analog and digital functionality on the same chip and are difficult to simulate with conventional CAD software, such as SPICE or Simulink, for three main reasons: (1) they are described by a large number of equations; (2) the equations involve highly discontinuous non-linear operations at the clock period of the digital circuit; and (3) the equations are currently solved using slow, time-marching, algorithms (Runge-Kutta type).
Recently, several approaches to fast simulation of mixed-signal circuits have been presented. Several are listed in the set of references below and are described here. Opal et al. presented a basic approach for circuits with a clock period in which linear differential equations are solved by one matrix multiply per clock cycle. In their method, a strong nonlinearity, such as the quantizer in a delta-sigma modulator, is simulated with a behavioral model at each clock period. Schreier and Zhang use a similar approach to construct recursion relations that update state variables of a delta-sigma modulator from time t to time t+Tc, where Tc represents the clock period. Cherry and Snelgrove compare three approaches: the recursion relation or direct integration approach, the time-marching method, and a z-domain extraction procedure, which were intended to combine the speed of the recurrence relations with the versatility of the time-marching method. Zhou et al., and Meliopoulos and Lee, have considered wavelet methods for use in general nonlinear circuit simulation and transient analysis.
There are several problems with the above-mentioned techniques. The matrix operator and direct integration methods are not sufficiently general to solve realistic circuit problems. The matrix in the matrix method is difficult to derive and is not small or sparse in general. The time-marching method is slow. The z-domain extraction procedure is difficult to generalize for real circuits. The wavelet techniques considered by Zhou et al., and Meliopoulos and Lee, could not be applied to mixed-signal circuits because of the number of wavelets required for time simulation over an entire time interval of 214 or more clock periods needed for analyzing mixed-signal circuits.
Accordingly, there exists a need in the art for a technique that overcomes the aforementioned limitations and that permits the use of wavelet operators to simulate the performance of systems in general, and mixed-signal circuits in particular.
The following references are provided for additional, non-critical, information that may be of use to the reader (all are hereby incorporated herein by reference in their entirety):    [1] Y. Ahmed and A. Opal, “An efficient simulation method for oversampled delta-sigma modulators,” Proceedings of the 37th Midwest Symposium on Circuits and Systems, vol. 2, 1994, pp. 1164-1167.    [2] A. Opal, “Sampled data simulation of linear and nonlinear circuits,” IEEE Trans. CAD Integrated Circuits and Syst., vol. 15, no. 3, March 1996, pp. 295-307.    [3] Y. Dong and A. Opal, “Time-domain thermal noise simulation of switched capacitor circuits and delta-sigma modulators,” IEEE Trans. CAD Integrated Circuits and Syst., vol. 19, no. 4, April 2000, pp. 473-481.    [4] R. Schreier and B. Zhang, “Delta-Sigma modulators employing continuous-time circuitry,” IEEE Trans. Circuits and Syst. I, vol. 43, no. 4, April 1996, pp. 324-332.    [5] J. A. Cherry and W. M. Snelgrove, “Approaches to simulating continuous-time delta-sigma modulators,” Proceedings of the 1998 IEEE International Symposium on Circuits and Systems, ISCAS '98, vol. 1, 1998, pp. 587-590.    [6] D. Zhou, W. Cai, and Wu Zhang, “An adaptive wavelet method for nonlinear circuit simulation,” IEEE Trans. Circuits and Syst. I, vol. 46, no. 8, August 1999, pp. 931-938.    [7] A. P. S. Meliopoulos and C.-H. Lee, “An alternative method for transient analysis via wavelets,” IEEE Trans. Power Delivery, vol. 15, no. 1, January 2000, pp. 114-121. A. W. Galli, “Discussion of ‘An alternative method for transient analysis via wavelets,’ ” Ibid, no. 4, October 2000, p. 1326. A. P. S. Meliopoulos and C.-H. Lee, “Closure to discussion of ‘An alternative method for transient analysis via wavelets,’ ” Ibid, no. 4, October 2000, pp. 1326-1327.    [8] M. Unser and A. Aldroubi, Polynomial Splines and Wavelets—A Signal Processing Perspective, in Wavelets: A Tutorial in Theory and Applications, ed. by C. K. Chui, Academic Press, New York, 1992, pp. 91-122.    [9] H. L. Resnikoff and R. O. Wells, Jr., Wavelet Analysis, Springer, N.Y., 1998, pp. 236-265, 281-340.    [10] G. Raghavan, J. F. Jensen, J. Laskowski, M. Kardos, M. G. Case, M. Sokolich, and S. Thomas III, “Architecture, design, and test of continuous-time tunable intermediate-frequency bandpass delta-sigma modulators,” IEEE J. Solid-State Circuits, vol. 36, no. 1, January 2001, pp. 5-13.    [11] T. Kaplan, P. Petre and G. C. Valley, “State variable simulation of non-idealities in a continuous time delta-sigma modulator,” submitted to IEEE J. CAD Integrated Circuits and Syst., May 2002.    [12] W. H. Press, S. A Teukolosky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran, the Art of Scientific Computing, Second Edition, Cambridge University Press, 1992pp. 340-386.    [13] O. C. Zienkiewicz and R. L. Taylor, Finite Element Method: Volume 1 The Basis, Butterworth and Heinemann, Oxford, 2000.    [14] G. Beylkin, “On the representation of operators in bases of compactly supported wavelets,”, SIAM Journal of Numerical analysis, vol. 6, no. 6, December 1992, pp. 1716-1740.